31.3. CURVILINEAR COORDINATES 617

31.3 Curvilinear CoordinatesThere are many ways to identify a point in n dimensional space with an ordered list of realnumbers. Some of these are spherical coordinates, cylindrical coordinates and rectangu-lar coordinates and these particular examples are discussed earlier. I will denote by y therectangular coordinates of a point in n dimensional space which I will go on writing as Rn.Thus y =

(y1 · · · yn

). It follows there are equations which relate the rectangular co-

ordinates to some other coordinates(

x1 · · · xn). In spherical coordinates, these were

ρ,φ ,θ where the geometric meaning of these were described earlier. However, completelygeneral systems are to be considered here, with certain stipulations. The idea is

yk = yk (x1, ...,xn) , y = y(x1, ...,xn)

Let(

x1 · · · xn)∈ D ⊆ Rn be an open set and let

x→ y(x1, ...,xn)≡M

(x1, ...,xn)

satisfyM is C2, (31.13)

M is one to one. (31.14)

Letting x ∈ D, we can writeM (x) = Mk (x)ik

where, as usual, ik are the standard basis vectors for Rn, ik being the vector in Rn whichhas a one in the kth coordinate and a 0 in every other spot. Thus yk = Mk (x) where this yk

refers to the kth rectangular coordinate of the point y as just described.For a fixed x ∈ D, we can consider the space curves,

t →M (x+ tik)≡ y (x+ tik)

for t ∈ I, some open interval containing 0. Then for the point x,we let

ek ≡∂M

∂xk (x)≡ ddt

(M (x+ tik)) |t=0 ≡∂y

∂xk (x)

Denote this vector as ek (x) to emphasize its dependence on x. The following pictureillustrates the situation in R3.

e1e2e3

t →M(x10,x

20, t)

t →M(t,x20,x

30)

t →M(x10, t,x

30)